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Module C7 – Total change – an introduction to integral calculus Module C7 Total change – an introduction to integral calculus 7 Table of Contents Introduction .................................................................................................................... 7.1 7.1 Area under the curve .............................................................................................. 7.3 7.2 The definite integral ............................................................................................... 7.11 7.3 The antiderivative .................................................................................................. 7.19 7.4 Steps in integration ................................................................................................ 7.19 7.4.1 Using standard rules of integration .................................................................. 7.19 7.4.2 Integrals of functions with constant multiples ................................................. 7.21 7.4.3 Integrals of sum and difference functions ........................................................ 7.21 7.5 More areas ............................................................................................................. 7.22 7.6 Applications of integral calculus ........................................................................... 7.31 7.7 A taste of things to come ....................................................................................... 7.36 7.8 Post-test ................................................................................................................. 7.37 7.9 Solutions ................................................................................................................ 7.39 Module C7 – Total change – an introduction to integral calculus 7.1 Introduction In this final section we will investigate another important aspect of calculus called integral calculus. One of the most important applications of integral calculus is to find areas and volumes. These problems have been considered by practitioners since the earliest times with the Greek mathematician Eudoxus (about 400 BC) credited with developing one of the first steps in calculus. He found an approximate area of the circle by sandwiching it between polygons with more and more sides. This was developed further by Archimedes to include many types of shapes, but it wasn’t until the 17th Century when Johannes Kepler, during the course of his astronomical investigations, wanted a method for finding areas of sectors that ideas moved ahead. Kepler was interested in two things. Firstly, as an astronomer, he wanted to find the area of an ellipse formed by the path of planets around the sun. He approximated this by using areas of triangles with their vertex at the sun. Planet Sun Secondly, as a connoisseur in wine, so the story goes, he wanted to find the exact method to calculate the volume of wine in kegs. Kepler thought of the wine barrel as being made up of infinitely many infinitely thin disks, the sum of whose areas became the volume. From a more every day perspective, suppose you were a wine merchant in the 17th Century. In those days the kegs were all hand made, and each one would hold a different amount of wine. You would need to know approximately how much wine you were selling to the innkeepers. Just as important the innkeepers would want to keep a check on you to ensure you were not overcharging on the amount of wine bought. 7.2 TPP7183 – Mathematics tertiary preparation level C In the diagram below, think about the ways you could estimate the volume of wine in the keg, without pouring out the wine litre by litre, or by submerging the keg in a larger container of water. Using the ideas of Archimedes and Kepler, one way you could do it is by thinking of the keg as a cylinder – taking the maximum and minimum volume of cylinders and averaging them. Another way of thinking about it is to think of ‘slicing’ the keg into a number of cylinders and then adding them up; the more cylinders you took the more accurate you would be.
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